It is a theorem of A. Levy, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1} V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ sentences.

One might expect that the "amount" of elementarity will grow quickly as we progress with large cardinal axioms, however for the next step, $V_\kappa\prec_{\Sigma_2}V$ we need to get much higher. In order to assure this level of elementarity a supercompact is enough (is it too strong? judging by the stage this theorem appears in Jech's and Kanamori's textbooks I would say that if it is too strong then it is not strong by that much)

To have $\Sigma_3$ we need to go even further to extendible cardinals (again, this might be too strong. I am not too familiar with this notion yet).

Is there a known large cardinal notion to give $\Sigma_4$ elementarity of $V_\kappa$? What about larger $n$?

I would expect complete elementarity to fail due to some Kunen inconsistency theorem sort of argument, is this true?

Are there results in the reverse direction? Namely if $\kappa$ is such that $V_\kappa\prec_{\Sigma_k}V$ then $\kappa$ has to be inaccessible/supercompact/extendible/etc

If we use all sort of set theoretic notions to measure how far $V$ is from an inner model (forcing axioms, large cardinals, how the cardinals behave in the inner model compared to $V$, sharps and covering theorems, etc etc).

Assuming the answer to the first question is not "It is inconsistent.", is there a useful way to use this approach to measure the difference between $V$ and its inner models?

3 Answers
3

The hypothesis that $V_\kappa$ is $\Sigma_k$ elementary or
even fully elementary in $V$ is much weaker than you say.

One can see part of this quite easily by observing that for
any inaccessible cardinal $\delta$, then
$V_\delta\models\text{ZFC}$ and there are a club of
ordinals $\alpha$ with $V_\alpha\prec V_\delta$. In
particular, if $\delta$ is Mahlo, then there are a
stationary set of inaccessible cardinals $\kappa$ with
$V_\kappa$ fully elementary in $V_\delta$.

In particular, if we lived inside $V_\delta$, we would
believe that there is a stationary proper class of
inaccessible cardinals $\kappa$ with $V_\kappa$ as fully
elementary in the universe as desired.

It turns out that although we can express
$V_\kappa\prec_{\Sigma_k} V$ as a first-order assertion of
$\kappa$ and $k$, it is not possible to express full
elementary $V_\kappa\prec V$ as a single first-order
assertion of set theory. Instead, we may use a scheme.

Thus, we introduce $\kappa$ as a constant symbol, and
consider the scheme, denoted "$V_\kappa\prec V$ ", asserting
of every formula $\varphi$ that $$\forall x\in V_\kappa\
(\varphi(x) \iff V_\kappa\models\varphi[x]\ ).$$ If we
add the assumption that $\kappa$ is inaccessible, then this
is known as the Levy scheme.

"ORD is Mahlo". That is, the scheme asserting of every
definable (with parameters) proper class club, that it
contains an inaccessible cardinal.

Proof. The first implies that $V_\kappa$ satisfies ORD is
Mahlo, since $\kappa$ will be a limit point and hence an
element of any such club as defined in $V$ using parameters
below $\kappa$. If the second is consistent, then so is the
first by a compactness argument, using the reflection
theorem. QED

Meanwhile, if you drop the inaccessibility requirement,
then you can attain the following, which many set theorists
find surprising.

Proof. If ZFC is consistent, then so is every finite
fragment of the scheme $V_\kappa\prec V$, by the reflection
theorem. QED

One can even attain a proper class club
$C\subset\text{ORD}$ of cardinals, with each $\kappa\in C$
satisfying the scheme $V_\kappa\prec V$, without going
beyond ZFC in consistency strength.

Both versions of the axiom $V_\kappa\prec V$ were important
in my paper on the maximality
principle, the
principle asserting that any statement that is forceable in
such a way that it remains true in all further extensions
is already true. It turned out that one can force the
maximality principle only from a model of $V_\kappa\prec V$
(and you need $\kappa$ inaccessible for the boldface
maximality principle).

Oh, after posting I find now an abundance of expert answers!
–
Joel David HamkinsJul 28 '11 at 23:32

2

It was recently suggested to me to ask these sort of questions here rather than on math.SE :-)
–
Asaf KaragilaJul 28 '11 at 23:35

(Since MO has a crappy comments notification): So the theorem "If $\kappa$ is X then $V_\kappa\prec_{\Sigma_n}V$" is in fact a metatheorem?
–
Asaf KaragilaJul 31 '11 at 8:34

I'm not sure I follow your question. For any fixed finite k, then we can assert $V_\kappa\prec_{\Sigma_k} V$ by a single assertion. There are a closed unbounded class of such $\kappa$, provided by the reflection theorem, and so under Ord is Mahlo, there are many inaccessible such $\kappa$. This hypothesis is weaker in consistency strength than a Mahlo cardinal. For fixed k, you don't need full Ord is Mahlo, but only Ord is Mahlo for certain $\Sigma_n$ definable class clubs, which is a single assertion, and you can figure out $n$ from $k$.
–
Joel David HamkinsJul 31 '11 at 16:51

This following result answers the third bullet item question in the negative.

Proposition. Suppose $(M,\in)$ is a transitive model of $ZF$ of uncountable cofinality. Then there is some ordinal $\alpha$ in $M$ of countable cofinality such that $(V_{\alpha})^M$ is a full elementary submodel of $M$.

Proof: Use the reflection theorem to produce an increasing sequence $\alpha_k$ for each $k \in \omega$ such that $(V_{\alpha_k})^M$ is a $\Sigma_k$-elementary submodel of $M$. The desired $\alpha$ is the union of the $\alpha_k$'s. QED

So it is quite possible to have $\kappa$ such that $V_\kappa$ is a full elementary submodel of $V$, without $\kappa$ being even regular, let alone inacessible.

The second and third of the bulleted questions are answered by an old theorem of Montague and Vaught. Suppose $\mu$ is the first inaccessible cardinal. Then there is $\kappa<\mu$ such that $V_\kappa\prec V_\mu$. Thus, from the point of view of $V_\mu$, there is an elementary submodel of the universe of the form $V_\kappa$, even though there is no inaccessible cardinal.

@Andreas: Many thanks for the very quick reply. In your answer you write about downwards reflection, that is in $V$ we go down from an inaccessible, and then chop off the "end" of the universe (above $\mu$, that is), and we are done. Does $V_\mu$ know internally that $V_\kappa$ is an elementary submodel?
–
Asaf KaragilaJul 28 '11 at 23:55

Asaf, the concept of "being an elementary submodel of the universe" is not expressible by a single assertion of set theory (if it were, you could get a contradiction in these models by letting $\kappa$ be least such, but then $V_\kappa$ would have also to have one, contradiction). But for any finite level of elementarity, $V_\mu$ can indeed observe that $V_\kappa$ is $\Sigma_n$-elementarity, since the satisfaction of statements in $V_\kappa$ is absolute between $V_\mu$ and $V$.
–
Joel David HamkinsJul 29 '11 at 0:04